Exploratory Data Analysis
\(\hspace{0.3cm}\) More articles: \(\hspace{0.1cm}\) Estadistica4all
\(\hspace{0.3cm}\) Author: \(\hspace{0.1cm}\) Fabio Scielzo Ortiz
\(\hspace{0.3cm}\) If you use this article, please, reference it:
\(\hspace{0.5cm}\) Scielzo Ortiz, Fabio. (2023). Exploratory Data Analysis. http://estadistica4all.com/Articulos/EDA.html
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1 Exploratory Data Analysis (EDA)
Exploratory data analysis (EDA) refers to the descriptive statistical analysis of a data-set.
Next we are going to propose a methodology to carry out an EDA, using Python as programming lenguage.
2 First step: data-set structure
2.1 Import data-set
First of all, we import the data-set with which we will work.
import pandas as pd
Netflix_Data = pd.read_csv('titles.csv')Netflix_Data| id | title | type | description | release_year | age_certification | runtime | genres | production_countries | seasons | imdb_id | imdb_score | imdb_votes | tmdb_popularity | tmdb_score | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | ts300399 | Five Came Back: The Reference Films | SHOW | This collection includes 12 World War II-era p… | 1945.0 | TV-MA | 51 | [‘documentation’] | [‘US’] | 1.0 | NaN | NaN | NaN | 0.600 | NaN |
| 1 | tm84618 | Taxi Driver | MOVIE | A mentally unstable Vietnam War veteran works … | 1976.0 | R | 114 | [‘drama’, ‘crime’] | [‘US’] | NaN | tt0075314 | 8.2 | 808582.0 | 40.965 | 8.179 |
| 2 | tm154986 | Deliverance | MOVIE | Intent on seeing the Cahulawassee River before… | 1972.0 | R | 109 | [‘drama’, ‘action’, ‘thriller’, ‘european’] | [‘US’] | NaN | tt0068473 | 7.7 | 107673.0 | 10.010 | 7.300 |
| 3 | tm127384 | Monty Python and the Holy Grail | MOVIE | King Arthur, accompanied by his squire, recrui… | 1975.0 | PG | 91 | [‘fantasy’, ‘action’, ‘comedy’] | [‘GB’] | NaN | tt0071853 | 8.2 | 534486.0 | 15.461 | 7.811 |
| 4 | tm120801 | The Dirty Dozen | MOVIE | 12 American military prisoners in World War II… | 1967.0 | NaN | 150 | [‘war’, ‘action’] | [‘GB’, ‘US’] | NaN | tt0061578 | 7.7 | 72662.0 | 20.398 | 7.600 |
| … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … |
| 5845 | tm1014599 | Fine Wine | MOVIE | A beautiful love story that can happen between… | 2021.0 | NaN | 100 | [‘romance’, ‘drama’] | [‘NG’] | NaN | tt13857480 | 6.8 | 45.0 | 1.466 | NaN |
| 5846 | tm898842 | C/O Kaadhal | MOVIE | A heart warming film that explores the concept… | 2021.0 | NaN | 134 | [‘drama’] | [] | NaN | tt11803618 | 7.7 | 348.0 | NaN | NaN |
| 5847 | tm1059008 | Lokillo | MOVIE | A controversial TV host and comedian who has b… | 2021.0 | NaN | 90 | [‘comedy’] | [‘CO’] | NaN | tt14585902 | 3.8 | 68.0 | 26.005 | 6.300 |
| 5848 | tm1035612 | Dad Stop Embarrassing Me - The Afterparty | MOVIE | Jamie Foxx, David Alan Grier and more from the… | 2021.0 | PG-13 | 37 | [] | [‘US’] | NaN | NaN | NaN | NaN | 1.296 | 10.000 |
| 5849 | ts271048 | Mighty Little Bheem: Kite Festival | SHOW | With winter behind them, Bheem and his townspe… | 2021.0 | NaN | 7 | [‘family’, ‘animation’, ‘comedy’] | [] | 1.0 | tt13711094 | 7.8 | 18.0 | 2.289 | 10.000 |
5850 rows × 15 columns
2.2 Data-set conceptual description
This data-set has information about 15 variables on 5850 Netflix titles.
Next table has a brief conceptual description about data-set variables:
| Variable | Descripción | Tipo |
|---|---|---|
| id | The title ID on JustWatch | Identifier |
| title | The name of the title | Text |
| type | TV show or movie | Categorical |
| description | A brief description | Text |
| release_year | release year | Quantitative |
| age_certification | age rating | Categorical |
| runtime | the number of episodes (show), the duration time in minutes (movie) | Quantitative |
| genres | A list of genres | Categorical |
| production_countries | A list of countries that produced the title | Categorical |
| seasons | Number of seasons if it’s a SHOW | Quantitative |
| imdb_id | The title ID on IMDB | Identifier |
| imdb_score | Rating on IMDB | Quantitative |
| imdb_votes | number of votes on IMDB | Quantitative |
| tmdb_popularity | Popularity on TMDB | Quantitative |
| tmdb_score | Rating on TMDB | Quantitative |
2.3 Data-set size
We can get the data-set size as the number of rows and columns of the data-set.
Netflix_Data.shape(5850, 15)As discussed above, the data-set has 5850 rows and 15 columns.
2.4 info() method
Netflix_Data.info()<class 'pandas.core.frame.DataFrame'>
RangeIndex: 5850 entries, 0 to 5849
Data columns (total 15 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 id 5850 non-null object
1 title 5849 non-null object
2 type 5850 non-null object
3 description 5832 non-null object
4 release_year 5850 non-null int64
5 age_certification 3231 non-null object
6 runtime 5850 non-null int64
7 genres 5850 non-null object
8 production_countries 5850 non-null object
9 seasons 2106 non-null float64
10 imdb_id 5447 non-null object
11 imdb_score 5368 non-null float64
12 imdb_votes 5352 non-null float64
13 tmdb_popularity 5759 non-null float64
14 tmdb_score 5539 non-null float64
dtypes: float64(5), int64(2), object(8)
memory usage: 685.7+ KBinfo() method give us column names, number of non null values in each column and column type.
2.5 Column types
Netflix_Data.dtypesid object
title object
type object
description object
release_year int64
age_certification object
runtime int64
genres object
production_countries object
seasons float64
imdb_id object
imdb_score float64
imdb_votes float64
tmdb_popularity float64
tmdb_score float64
dtype: objectIt’s another way to get column types.
Object is the typical type of categorical variables, identifier or text.
Float64 and int64 is the typical type of quantitative variables, float64 for continuous one, and int64 for discrete one.
2.6 Change column types
We can change the type of a column with astype() method:
Netflix_Data['release_year'] = Netflix_Data['release_year'].astype('float')We can check if the changes have been done correctly:
Netflix_Data.dtypesid object
title object
type object
description object
release_year float64
age_certification object
runtime int64
genres object
production_countries object
seasons float64
imdb_id object
imdb_score float64
imdb_votes float64
tmdb_popularity float64
tmdb_score float64
dtype: object2.7 NaN identification
A NaN is a not a number value. NaN is equivalent to missing value.
We are going to calculate, for each variable, the proportion of missing values over the total number of observations:
Prop_NA = Netflix_Data.isnull().sum() / len(Netflix_Data)
Prop_NAid 0.000000
title 0.000171
type 0.000000
description 0.003077
release_year 0.000000
age_certification 0.447692
runtime 0.000000
genres 0.000000
production_countries 0.000000
seasons 0.640000
imdb_id 0.068889
imdb_score 0.082393
imdb_votes 0.085128
tmdb_popularity 0.015556
tmdb_score 0.053162
dtype: float64We can see that there are variables with a high proportion of missing values, as age_certification (44.77%).
season would be the variable with higher proportion of missing values, but it is because of season only is defined for type=SHOW.
3 Auxiliar step: data preprocessing
3.1 Categorization of cuantitative variables
3.2 Dummification of categorical variables
3.3 Recodefication of categorical variables
3.4 Variable Scaling
4 Second step: statistical description
4.1 Statistical variable
A statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\) can be modeled as a random variable.
Under this approach, we can apply all probability theory on random variables to statistical variables. \(\\[0.4cm]\)
4.2 Range of a statistical variable
The range of a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\) is denoted by \(\hspace{0.05cm}Range(\mathcal{X}_k)\hspace{0.05cm}\), and is defined as the set of possible values of \(\hspace{0.05cm}\mathcal{X}_k\). \(\\[0.4cm]\)
4.2.1 Statistical variable types: quantitative and categorical
The variable \(\mathcal{X}_k\) is quantitative if the elements of it´s range are conceptually numbers. \(\\[0.5cm]\)
The variable \(\mathcal{X}_k\) is categorical if the elements of it´s range aree labels or categories (they can be numbers at a symbolic level but not at a conceptual level). \(\\[0.4cm]\)
4.2.2 Quantitative variable types: continuous and discrete
We can distinguish at least two types of quantitative variables: continuous and discrete.
\(\mathcal{X}_k\hspace{0.05cm}\) is continuous if \(\hspace{0.05cm}Range(\mathcal{X}_k)\hspace{0.05cm}\) is a not countable set. \(\\[0.5cm]\)
\(\mathcal{X}_k\hspace{0.05cm}\) is discrete if \(\hspace{0.05cm}Range(\mathcal{X}_k)\hspace{0.05cm}\) is countable set. \(\\[0.2cm]\)
Note:
In particular, variables whose range is a finite set will be discrete.
Variables whose range isn´t a finite set will be continuous. \(\\[0.4cm]\)
4.2.3 Categorical variable types: r-ary
Let \(\mathcal{X}_k\) a categorical variable ,
- \(\mathcal{X}_k\) is r-aria if it´s range has r elements that are categories or labels.
In Statistics binary (2-aria) categorical variables are particularly important. \(\\[0.4cm]\)
4.2.4 Categorical variable types: nominal and ordinal
Let \(\mathcal{X}_k\) a \(r\)-ary categorical variable.
\(\mathcal{X}_k\) is nominal if there is no ordering between the \(r\) categories of it’s range. \(\\[0.4cm]\)
\(\mathcal{X}_k\) is ordinal if there is ordering between the \(r\) categories of it’s range. \(\\[0.4cm]\)
4.3 Sample of a statistical variable
Given a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\).
A sample of \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\) is a vector of values of \(\hspace{0.05cm}\mathcal{X}_k\), called observations.
Therefore:
\[ X_k \hspace{0.05cm} = \hspace{0.05cm} \begin{pmatrix} x_{1k} \\ x_{2k}\\ ... \\ x_{nk} \end{pmatrix} \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t \\ \]
is a sample of a statistical variable because is a vector with the values or observations of the variable \(\hspace{0.05cm} \mathcal{X}_k \hspace{0.05cm}\) for \(\hspace{0.05cm} n \hspace{0.05cm}\) elements or individuals of a sample.
Where: \(\hspace{0.1cm} x_{ik}\hspace{0.05cm}\) is the value \(\hspace{0.05cm} i\)-th observation of the variable \(\hspace{0.05cm} \mathcal{X}_k\). \(\\[0.4cm]\)
4.4 Arithmetic Mean
Given a quantitative statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The arithmetic mean of \(\hspace{0.05cm}X_k \hspace{0.05cm}\) is defined as: \(\\[0.3cm]\)
\[\overline{\hspace{0.05cm} X_k \hspace{0.05cm} } \hspace{0.1cm}=\hspace{0.1cm} \dfrac{1}{n} \cdot \sum_{i=1}^n \hspace{0.05cm} x_{ik}\] \(\\[0.4cm]\)
4.5 Weighted Mean
Given a quantitative statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
And given a weights for each observation of the variable \(\hspace{0.05cm} \mathcal{X}_k \hspace{0.2cm} \Rightarrow \hspace{0.2cm}\) \(w \hspace{0.05cm} = \hspace{0.05cm} (w_1,w_2,...,w_n)^t\)
The weighted mean of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) with the weights vector \(\hspace{0.05cm} w \hspace{0.05cm}\) is defined as:
\[ \overline{X_k} (w) \hspace{0.1cm} = \hspace{0.1cm} \dfrac{1}{\hspace{0.1cm}\sum_{i=1}^{n} \hspace{0.05cm} w_{i} \hspace{0.1cm}} \hspace{0.05cm}\cdot\hspace{0.05cm} \sum_{i=1}^{n} \hspace{0.1cm} x_{ik} \cdot w_i \] \(\\[0.4cm]\)
4.6 Geometric Mean
Given the variable \(\hspace{0.05cm} X_k=(x_{1k}, x_{2k},...,x_{nk})^t\).
The geometric mean of the variable \(\hspace{0.05cm}X_k\hspace{0.05cm}\) is defined as: \(\\[0.3cm]\)
\[ \overline{X_k}_{geo} \hspace{0.05cm} = \hspace{0.05cm} \sqrt{\Pi_{i=1}^{n} x_{ik}} \hspace{0.05cm} = \hspace{0.05cm} \sqrt{x_{1k}\cdot x_{2k}\cdot...\cdot x_{nk}} \] \(\\[0.4cm]\)
4.7 Median
Given a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The median of \(\hspace{0.05cm}X_k \hspace{0.05cm}\) is defined as a value \(Me(X_k)\) such that: \(\\[0.3cm]\)
\[\dfrac{1}{n} \cdot \sum_{i=1}^n \hspace{0.1cm} \mathbb{I} \hspace{0.05cm} \bigl[ \hspace{0.1cm} x_{ik} \hspace{0.05cm} \leq \hspace{0.05cm} Me(X_k) \hspace{0.1cm} \bigr] \hspace{0.1cm} = \hspace{0.1cm} 0.50\]
where: \(\hspace{0.15cm}\mathbb{I}\hspace{0.1cm}\) is the indicator function. \(\\[0.4cm]\)
4.8 Mode
Given a categorical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The mode of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is the most repeated value in \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\), so, the mode of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is the most frequent value of \(\hspace{0.05cm} X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) \(\\[0.4cm]\)
4.9 Variance
Given a quantitative variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The variance of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as:
\[\sigma(X_k)^2 \hspace{0.1cm} = \hspace{0.1cm} \dfrac{1}{n} \cdot \sum_{i=1}^n \hspace{0.05cm} \left(\hspace{0.05cm} x_{ik} - \overline{X_k} \hspace{0.05cm}\right)^2\]
The standard deviation or standard error of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as:
\[\sqrt{ \sigma(X_k)^2 } \hspace{0.1cm} = \hspace{0.1cm} \dfrac{1}{n} \cdot \sum_{i=1}^n \left( \hspace{0.05cm} x_{ik} - \overline{X_k} \hspace{0.05cm} \right)\] \(\\[0.4cm]\)
4.10 Median Absolute Deviation
Given a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The median absolute deviation (MAD) of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as:
\[MAD(X_k) \hspace{0.1cm} = \hspace{0.1cm} Me \bigl( \hspace{0.1cm} \left| \hspace{0.05cm} X_k - Me(X_k) \hspace{0.05cm} \right| \hspace{0.1cm} \bigr) \hspace{0.1cm} = \hspace{0.1cm} Me \hspace{0.1cm} \Bigr[ \hspace{0.1cm} \left( \hspace{0.2cm} \left| \hspace{0.1cm} x_{ik} - Me(X_k) \hspace{0.1cm} \right| \hspace{0.15cm} : \hspace{0.15cm} i = 1,\dots,n \hspace{0.2cm} \right) \hspace{0.1cm} \Bigr]\] \(\\[0.4cm]\)
4.11 Quantiles
Given a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The \(\hspace{0.05cm}q\)-order quantile of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as a value \(Q(X_k , q)\) such that:
\[\dfrac{1}{n} \cdot \sum_{i=1}^n \hspace{0.1cm} \mathbb{I} \hspace{0.05cm} \bigl[ \hspace{0.1cm} x_{ik} \hspace{0.05cm} \leq \hspace{0.05cm} Q(\hspace{0.05cm} X_k \hspace{0.05cm},\hspace{0.05cm} q \hspace{0.05cm}) \hspace{0.1cm} \bigr] \hspace{0.1cm} = \hspace{0.1cm} q\]
where: \(\hspace{0.15cm}\mathbb{I}\hspace{0.1cm}\) is the indicator function. \(\\[0.3cm]\)
Observation:
The median is the 0.5-order quantile. \(\\[0.4cm]\)
4.12 Kurtosis
Given a quantitative variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The kurtosis coefficient of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as: \(\\[0.35cm]\)
\[ \Psi(X_k) = \dfrac{\mu_{4}}{\sigma(X_k)^{4}} \]
where:
\[ \mu_{4}\hspace{0.1cm} =\hspace{0.1cm} \frac{1}{n} \cdot \sum_{i=1}^{n} \hspace{0.05cm} x_{ik}^4 \\[0.3cm] \]
Propierties:
If \(\hspace{0.12cm}\Psi(X_k) \hspace{0.05cm} > \hspace{0.05cm} 3\hspace{0.08cm}\) \(\hspace{0.2cm}\Rightarrow\hspace{0.2cm}\) the distribution of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is more pointed and with longer tails than the normal distribution. \(\\[0.5cm]\)
If \(\hspace{0.12cm}\Psi(X_k) \hspace{0.05cm} < \hspace{0.05cm} 3\hspace{0.08cm}\) \(\hspace{0.2cm}\Rightarrow\hspace{0.2cm}\) the distribution of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is less pointed and with shorter tails than the normal distribution. \(\\[0.4cm]\)
4.13 Skewness
Given a quantitative variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
The skewness coefficient of \(\hspace{0.05cm} X_k \hspace{0.05cm}\) is defined as: \(\\[0.25cm]\)
\[ \Gamma(X_k) = \dfrac{\mu_{3}}{\sigma(X_k)^{3}} \]
where:
\[ \mu_{3}\hspace{0.1cm} =\hspace{0.1cm} \frac{1}{n} \cdot \sum_{i=1}^{n} \hspace{0.05cm} x_{ik}^3 \\[0.3cm] \]
Propierties:
Fisher’s skewness coefficient measures the degree of skewness in the distribution of a given statistical variable.
If \(\hspace{0.12cm} \Gamma(X_k) > 0\) \(\hspace{0.2cm} \Rightarrow \hspace{0.2cm}\) the distribution of \(X_k\) has skewness to the right. \(\\[0.6cm]\)
If \(\hspace{0.12cm} \Gamma(X_k) < 0\) \(\hspace{0.2cm} \Rightarrow \hspace{0.2cm}\) the distribution of \(X_k\) has skewness to the left. \(\\[0.4cm]\)
4.14 Outliers
There are several definitions of outlier, but here we are going to consider the classic one.
Given a statistical variable \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\), and a sample \(\hspace{0.05cm}X_k \hspace{0.05cm} = \hspace{0.05cm} \left( \hspace{0.01cm} x_{1k} \hspace{0.01cm} , \hspace{0.01cm}x_{2k}\hspace{0.01cm},\dots ,\hspace{0.01cm} x_{nk} \hspace{0.01cm}\right)^t\hspace{0.05cm}\) of that statistical variable.
For any \(\hspace{0.05cm} i\in \lbrace 1,...,n \rbrace\) ,
The observation \(\hspace{0.05cm} x_{ik}\hspace{0.05cm}\) of $ _k $ is an outlier if and only if:
\[x_{ik} \hspace{0.05cm} >\hspace{0.05cm} Q(X_k \hspace{0.05cm} , \hspace{0.05cm} 0.75) + 1.5\cdot IQR(X_k) \hspace{0.5cm}\text{or}\hspace{0.5cm} x_{ik} \hspace{0.05cm} <\hspace{0.05cm} Q(X_k \hspace{0.05cm} , \hspace{0.05cm} 0.25) - 1.5\cdot IQR(X_k) \\\]
where: \(\hspace{0.25cm} IQR(X_k) \hspace{0.12cm} = \hspace{0.12cm} Q(X_k \hspace{0.05cm} , \hspace{0.05cm} 0.75) \hspace{0.08cm} - \hspace{0.08cm} Q(X_k \hspace{0.05cm} , \hspace{0.05cm} 0.25) \hspace{0.25cm}\) is the interquartile range of \(\hspace{0.05cm} X_k \hspace{0.05cm}\).
4.15 Data Matrix
Given \(\hspace{0.05cm} p \hspace{0.05cm}\) statistical variables \(\hspace{0.05cm}\mathcal{X}_1, \mathcal{X}_2, \dots \mathcal{X}_p\hspace{0.05cm}\), and given a sample \(\hspace{0.05cm}X_k = (x_{1k},...,x_{nk})^t\hspace{0.05cm}\) of \(\hspace{0.05cm}\mathcal{X}_k\hspace{0.05cm}\) for each \(\hspace{0.05cm}k \in \lbrace 1,...,p \rbrace\).
A data matrix of the variables \(\hspace{0.05cm}\mathcal{X}_1,...,\mathcal{X}_1\hspace{0.05cm}\) would be: \(\\[0.35cm]\)
\[ X \hspace{0.05cm}=\hspace{0.05cm} \left( X_1 , X_2,\dots , X_p \right) \hspace{0.05cm}=\hspace{0.05cm} \begin{pmatrix} x_{1}^{t} \\ x_{2} ^t \\ ... \\ x_{n} ^t \end{pmatrix} \hspace{0.05cm}=\hspace{0.05cm} \begin{pmatrix} x_{11} & x_{12}&...&x_{1p}\\ x_{21} & x_{22}&...&x_{2p}\\ ...&...&...&...\\ x_{n1}& x_{n2}&...&x_{np} \end{pmatrix} \\ \]
where:
\(x_i ^t \hspace{0.05cm}=\hspace{0.05cm} \left( x_{i1}, x_{i2}, \dots , x_{ip} \right)\hspace{0.1cm}\) is the vector with the values of the \(\hspace{0.05cm} p \hspace{0.05cm}\) statistical variables \(\hspace{0.05cm}\mathcal{X}_1,\dots ,\mathcal{X}_p\hspace{0.05cm}\) for the \(\hspace{0.05cm}i\)-th element of the sample, for \(\hspace{0.05cm} i \in \lbrace 1,...,n \rbrace\) \(\\[0.4cm]\)
Observations:
\(X \hspace{0.05cm}=\hspace{0.05cm} \left( X_1 , X_2,\dots , X_p \right)\hspace{0.1cm}\) is a matrix with \(\hspace{0.05cm}p\hspace{0.05cm}\) columns and \(\hspace{0.05cm}n\hspace{0.05cm}\) rows, so, is a matrix of size \(\hspace{0.05cm} p\hspace{0.05cm} \text{x}\hspace{0.05cm}n\). \(\\[0.4cm]\)
4.16 Covariance
4.17 Correlation
4.18 Statistical description for quantitative variables
mean. median, variance, cuantiles, kurtosis, skewness, outliers
frequency tables –> https://www.statology.org/frequency-tables-python/
4.19 Statistical description for categorical variables
mode, quantiles
frequency tables
4.20 Statistical description for variable crossings (cruces de variables cuantis-categoricas, categroicas-categoricas, cuantis-cuantis)
quantitative-categorical –> mean, median, vaariance, quantiles etc BY GROUPS. Joint and conditional frequency tables.
categorical-categorical –> Joint and conditional frequency tables.
quantitative-quantitative –> transform to categorical-categorical case.
5 Third step: statistical visualization
5.1 Visualization for quantitative variables
5.2 Visualization for categorical variables
5.3 Visualization for quantitative-categorical
5.4 Visualization for categorical-categorical
6 Descripción Estadistica Básica
A continuación vamos a realizar una descripción estadistica básica de las variables, a traves de dicersos estadisticos básicos.
6.1 Estadisticos básicos para las variables cuantitativas
Para las variables cuantitativas:
Netflix_Data.describe()| release_year | runtime | seasons | imdb_score | imdb_votes | tmdb_popularity | tmdb_score | |
|---|---|---|---|---|---|---|---|
| count | 5850.000000 | 5850.000000 | 2106.000000 | 5368.000000 | 5.352000e+03 | 5759.000000 | 5539.000000 |
| mean | 2016.417094 | 76.888889 | 2.162868 | 6.510861 | 2.343938e+04 | 22.637925 | 6.829175 |
| std | 6.937726 | 39.002509 | 2.689041 | 1.163826 | 9.582047e+04 | 81.680263 | 1.170391 |
| min | 1945.000000 | 0.000000 | 1.000000 | 1.500000 | 5.000000e+00 | 0.009442 | 0.500000 |
| 25% | 2016.000000 | 44.000000 | 1.000000 | 5.800000 | 5.167500e+02 | 2.728500 | 6.100000 |
| 50% | 2018.000000 | 83.000000 | 1.000000 | 6.600000 | 2.233500e+03 | 6.821000 | 6.900000 |
| 75% | 2020.000000 | 104.000000 | 2.000000 | 7.300000 | 9.494000e+03 | 16.590000 | 7.537500 |
| max | 2022.000000 | 240.000000 | 42.000000 | 9.600000 | 2.294231e+06 | 2274.044000 | 10.000000 |
6.2 Estadisticos básicos para las variables categóricas
Para las variables categóricas (no cuantitativas, en general) :
Netflix_Data.loc[: , ['title', 'description', 'age_certification', 'genres', 'production_countries' ]].describe()| title | description | age_certification | genres | production_countries | |
|---|---|---|---|---|---|
| count | 5849 | 5832 | 3231 | 5850 | 5850 |
| unique | 5798 | 5829 | 11 | 1726 | 452 |
| top | The Gift | Five families struggle with the ups and downs … | TV-MA | [‘comedy’] | [‘US’] |
| freq | 3 | 2 | 883 | 484 | 1959 |
6.3 Gráficos conjuntos para las variables cuantitativas
En esta seccion vamos a hacer un analisis gráfico básico de las variables cuantitativas, consideradas de manera conjunta.
Cargamos las librerias necesarias para los gráficos:
import numpy as np
import seaborn as sns
import matplotlib as mpl
import matplotlib.pyplot as plt6.3.1 Histograma conjunto de las variables cuantitativas
Vamos a generar un grafico con un histograma para cada una de las variables cuantitativas:
fig, axs = plt.subplots(3, 3, figsize=(11, 11))
p1 = sns.histplot(data=Netflix_Data, x="release_year", stat="proportion", bins=15, color="skyblue", ax=axs[0, 0])
p2 = sns.histplot(data=Netflix_Data, x="runtime", stat="proportion", bins=15, color="olive", ax=axs[0, 1])
p2.axes.set(xlabel='runtime', ylabel=' ')
p3 = sns.histplot(data=Netflix_Data, x="seasons", stat="proportion", bins=15, color="blue", ax=axs[0, 2])
p3.axes.set(xlabel='seasons', ylabel=' ')
p4 = sns.histplot(data=Netflix_Data, x="imdb_score", stat="proportion", bins=15, color="teal", ax=axs[1, 0])
p4.axes.set(xlabel='imdb_score', ylabel=' ')
p5 = sns.histplot(data=Netflix_Data, x="imdb_votes", stat="proportion", bins=15, color="purple", ax=axs[1, 1])
p5.axes.set(xlabel='imdb_votes', ylabel=' ')
p6 = sns.histplot(data=Netflix_Data, x="tmdb_popularity", stat="proportion", bins=15, color="pink", ax=axs[1, 2])
p6.axes.set(xlabel='tmdb_popularity', ylabel=' ')
p7 = sns.histplot(data=Netflix_Data, x="tmdb_score", stat="proportion", bins=15, color="red", ax=axs[2, 0])
p7.axes.set(xlabel='tmdb_score', ylabel=' ')
fig.savefig('p1.png', format='png', dpi=1200)
plt.show()
Histograma conjunto de las variables cuantitativas
6.3.2 Box-Plot conjunto de las variables cuantitativas
Vamos a generar un grafico con un box-plot para cada una de las variables cuantitativas:
fig, axs = plt.subplots(3, 3, figsize=(11, 11))
p1 = sns.boxplot(data=Netflix_Data, x="release_year", color="skyblue", ax=axs[0, 0])
p2 = sns.boxplot(data=Netflix_Data, x="runtime", color="olive", ax=axs[0, 1])
p2.axes.set(xlabel='runtime', ylabel=' ')
p2.set_xticks( range(int(Netflix_Data['runtime'].min()) , int(Netflix_Data['runtime'].max()) , 100) )
p2.set_yticks( np.arange(0, 1, 0.1) )
p3 = sns.boxplot(data=Netflix_Data, x="seasons", color="blue", ax=axs[0, 2])
p3.axes.set(xlabel='seasons', ylabel=' ')
p4 = sns.boxplot(data=Netflix_Data, x="imdb_score", color="teal", ax=axs[1, 0])
p4.axes.set(xlabel='imdb_score', ylabel=' ')
p4.set_xticks( range(int(Netflix_Data['imdb_score'].min()) , int(Netflix_Data['imdb_score'].max()) , 300) )
p4.set_yticks( np.arange(0, 1, 0.1) )
p5 = sns.boxplot(data=Netflix_Data, x="imdb_votes", color="purple", ax=axs[1, 1])
p5.axes.set(xlabel='imdb_votes', ylabel=' ')
p5.set_xticks( range(int(Netflix_Data['imdb_votes'].min()) , int(Netflix_Data['imdb_votes'].max()/2) , 500000) )
p5.set_yticks( np.arange(0, 1, 0.1) )
p6 = sns.boxplot(data=Netflix_Data, x="tmdb_popularity", color="pink", ax=axs[1, 2])
p6.axes.set(xlabel='tmdb_popularity', ylabel=' ')
p6.set_xticks( range(int(Netflix_Data['tmdb_popularity'].min()) , int(Netflix_Data['tmdb_popularity'].max()+1) , 1000) )
p6.set_yticks( np.arange(0, 1, 0.1) )
p7 = sns.boxplot(data=Netflix_Data, x="tmdb_score", color="red", ax=axs[2, 0])
p7.axes.set(xlabel='tmdb_score', ylabel=' ')
p7.set_xticks( range(int(Netflix_Data['tmdb_score'].min()) , int(Netflix_Data['tmdb_score'].max()+1) , 2) )
p7.set_yticks( np.arange(0, 1, 0.1) )
plt.show()
Box-Plot conjunto de las variables cuantitativas
6.3.3 Empirical-Cumulative-Distribution-Function-Plot conjunto de las variables cuantitativas
Vamos a generar un grafico con un ECDF-plot para cada una de las variables cuantitativas:
fig, axs = plt.subplots(3, 3, figsize=(11, 11))
p1 = sns.ecdfplot(data=Netflix_Data, x="release_year", color="skyblue", ax=axs[0, 0])
p1.set_xticks( range(int(Netflix_Data['release_year'].min()) , int(Netflix_Data['release_year'].max()+20) , 20) )
p1.set_yticks( np.arange(0, 1, 0.1) )
p2 = sns.ecdfplot(data=Netflix_Data, x="runtime", color="olive", ax=axs[0, 1])
p2.axes.set(xlabel='runtime', ylabel=' ')
p2.set_xticks( range(int(Netflix_Data['runtime'].min()) , int(Netflix_Data['runtime'].max()) , 100) )
p2.set_yticks( np.arange(0, 1, 0.1) )
p3 = sns.ecdfplot(data=Netflix_Data, x="seasons", color="blue", ax=axs[0, 2])
p3.axes.set(xlabel='seasons', ylabel=' ')
p3.set_xticks( range(int(Netflix_Data['seasons'].min()) , int(Netflix_Data['seasons'].max()) , 4) )
p3.set_yticks( np.arange(0, 1, 0.1) )
p4 = sns.ecdfplot(data=Netflix_Data, x="imdb_score", color="teal", ax=axs[1, 0])
p4.axes.set(xlabel='imdb_score', ylabel=' ')
p4.set_xticks( range(int(Netflix_Data['imdb_score'].min()) , int(Netflix_Data['imdb_score'].max()) , 300) )
p4.set_yticks( np.arange(0, 1, 0.1) )
p5 = sns.ecdfplot(data=Netflix_Data, x="imdb_votes", color="purple", ax=axs[1, 1])
p5.axes.set(xlabel='imdb_votes', ylabel=' ')
p5.set_xticks( range(int(Netflix_Data['imdb_votes'].min()) , int(Netflix_Data['imdb_votes'].max()/2) , 500000) )
p5.set_yticks( np.arange(0, 1, 0.1) )
p6 = sns.ecdfplot(data=Netflix_Data, x="tmdb_popularity", color="pink", ax=axs[1, 2])
p6.axes.set(xlabel='tmdb_popularity', ylabel=' ')
p6.set_xticks( range(int(Netflix_Data['tmdb_popularity'].min()) , int(Netflix_Data['tmdb_popularity'].max()+1) , 1000) )
p6.set_yticks( np.arange(0, 1, 0.1) )
p7 = sns.ecdfplot(data=Netflix_Data, x="tmdb_score", color="red", ax=axs[2, 0])
p7.axes.set(xlabel='tmdb_score', ylabel=' ')
p7.set_xticks( range(int(Netflix_Data['tmdb_score'].min()) , int(Netflix_Data['tmdb_score'].max()+1) , 50) )
p7.set_yticks( np.arange(0, 1, 0.1) )
plt.show()
ECDF-Plot conjunto de las variables cuantitativas
6.4 Gráficos conjuntos para las variables categoricas
6.4.1 Bar-plot conjunto de las variables categóricas
Vamos a generar un grafico con un bar-plot para cada una de las variables categóricas, excepto para aquellas cuyo nº de categorias es excesivo, y por tanto no es práctico el gráfico:
fig, axs = plt.subplots(1, 2, figsize=(13, 6))
p1 = sns.countplot(x='type', data=Netflix_Data, ax=axs[0])
p1.set_xticklabels(['Movie', 'Show'])
p1.axes.set(xlabel='type', ylabel='count')
p2 = sns.countplot(x='age_certification', data=Netflix_Data, ax=axs[1])
plt.show()
Bar-Plot conjunto de variables categoricas
7 Análisis Estadístico
En la sección anterior se hizo una descripción estadistica básica de las variables del data-set con el que estamos trabajando, pero no se ha hecho ningun analisis de los resultados obtenidos.
En esta seccion además de ampliar la descripción estadistica de los datos, se llevará a cabo un analisis de los resultados obtenidos.